The nature of the COVID-19 pandemic may require governments to use privacy-encroaching technologies to help contain its spread. One technology involves co-location tracking through mobile Wi-Fi, GPS, and Bluetooth to permit health agencies to monitor people’s contact with each other, thereby triggering targeted social-distancing when a person turns out to be infected. The effectiveness of tracking relies on the willingness of the population to support such privacy encroaching measures. We report the results of two large surveys in the United Kingdom, conducted during the peak of the pandemic, that probe people’s attitudes towards various tracking technologies. The results show that by and large there is widespread acceptance for co-location tracking. Acceptance increases when the measures are explicitly time-limited and come with opt-out clauses or other assurances of privacy. Another possible future technology to control the pandemic involves “immunity passports”, which could be issued to people who carry antibodies for the COVID-19 virus, potentially implying that they are immune and therefore unable to spread the virus to other people. Immunity passports have been considered as a potential future step to manage the pandemic. We probe people’s attitudes towards immunity passports and find considerable support overall, although around 20% of the public strongly oppose passports.